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Image Transmission Under Arche Mirage Conditions by Waldemar H

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Image Transmission Under Arche Mirage Conditions by Waldemar H Image Transmission under Arche Mirage Conditions By Waldemar H. Lehn and H. Leonard Sawatzky' Summary: The arctic mirage, an optical effect fairly common in the high latitudes, occurs when the atmosphere exhibits qr eater-then-norrnul refractive capability. Light rays then follow paths concave toward th e earth, with curvature qr eater than or equal to that of the earth. The atmosphere acquires this refractive capability under conditions of temperature inversion, which exaggerate the fall-off of density (and hence refractive index) with elevation. Equations are presented that ca1culate density profile from any given temperature p rofi le : these results arc used to develop ru y path equations for nearly horizontal rays in the lower atmosphere. Ray paths are computed for two distinct mirage conditions (a strong sh al low inversion and a mild but deep inversion) as weIl as for th e standard atmosphere. With the aid of these paths, the appearance of th e environment is shown. The strong inversion produces the appearance of a seucer-shapcd earth (with image dtstortion near the horizon), whereas the mild inversion produces a "flat e arth" appearance that permits great viewing distances. Zusammenfassung: Die arktische Fata Morgana ist ein verhältnismäßig häufig vorkommender optischer Effekt in höheren Breiten, Er entsteht, wenn durch gesteigerte Densitätsschichtung der Atmosphäre Lichtstrahl­ bahnen gebrochen werden in einem Ausmaß, das der Krümmung der Erde gleicht oder sie überschreitet. Die Atmosphäre gewinnt diese Refraktionsfähigkeit unter Temperaturinversionsverhältnissen, welche die normale Densi tätsabnahrna mit zunehmender Höhe verstärken. Gleichungen, die das Densitätsprofil bei gegebenem Temperaturprofil errechnen lassen, werden vorgelegt. Aus den Ergebnissen lassen sich Behn-Gleidiunqen annähernd horizontaler Lichtstrahlen im bodennahen Raum entwickeln. Lichtstrahlbahnen werden als Beispiele berechnet und dargestellt, sowohl für eine flache, Jedoch starke, und eine weniger starke, aber tiefe Inversion, als auch bezüglich der normalen Atmosphäre. Mit Hilfe dieser Ergebnisse wird die optische Gestalt der Umwelt errechnet. Die starke Inversion ergibt den Aspekt einer schüsselförmigen Erde; die weniger starken den einer absolut flachen Erde. Beide Er­ scheinungen verursachen eine optische Ausdehnung des Blickfeldes bei weit zurÜckweichendem Horizont. 1. Introduclion The arctic mirage is an optical phenomenon that occurs fairly frequently in the high latitudes. The effect, also called "100m" or "superrefraction", occurs when the atmosphere refracts light rays more strongly than usual. Under these conditions ray paths become concave toward the earth, with radii of curvature less than or equal to the aarth's radius. Visual information can then be transmitted beyond the normal "horizon distance", and the surface takes on a flat or saucer-shaped appearance. This paper describes the phys­ ical basis for the mirage, and develops equations useful in predicting the appearance of the environment under mirage conditions. Even the normal atmosphere refracts rays mildly, due to the decreasing density (and hence refractive index) with altitude. The effect is accentuated to produce the arctic mirage when a warm air mass rests upon a cold surface. The conductive cooling from below develops a temperature inversion, with a steeper-than-normal density gradient. This gradient in turn produces the stronger refraction effects that create the arctic mirage. This refracting layer need not be in contact with the aarth's surface. The situation where warm air overlies cold air is equally effective, the necessary density gradient occuring in the boundary zone between the two layers. Hobbs reports that, in the high latitudes, the cold layer is frequently of the order of 1000 meters deep. It should be noted that the arctic mirage is fundamentally distinct from the fata morgana or desert mirage, which depends on the opposite physical effect, namely pronounced heating of the air in contact with the earths surface. The following sections develop the equations that permit calculation of ray paths when the temperature profile in the lower atmosphere is known. • Prof. Waldemar H. Lehn, Iteculty of Engineering, University of Mani toba, Winnipeg. Man. R3T 2N2 (Canada). Dr. H. Leonard Sawatzky, Department of Georgraphy, University of Manitoba, Winnipeg, Man. R3T 2N2 (Canada). 120 2. Ray paths in the refracting atmosphere The refractive effects of the atmosphere are weIl known. For visible light Bertram gives the refractive index as a function of density p (kq/rn'') by 6 n = 1 + (226 x 10- ) p (1) and the propagation velocity v by v = ein (2) where c is the velocity of light in vacuum. In the following analysis the atmosphere is considered to be laterally homogeneous, so that the only spatial variation in density comes with elevation z above the e ar ths surface. Attention will be concentrated on rays whose paths remain close to the surface; an analogous development can be used to analyse refractive zones at higher elevations. For such an atmosphere over a flat earth, Bertram derives an expression for the radius of curvature r at an arbitrary point on the ray: sin 8(z) dv (3) -=~ dz where O(z) is the angle between the ray and the vertical and v(z) is the velocity of propagation, at elevation z. From Sne lls Law of Refraction, the term (sin O)/v is a constant for any given ray. Coordinates are chosen such that the ray is contained in the xz plane. An additional set of coordinates (see Fig. 1) facilitates further calculations: a new abscissa u is chosen tangent to the ray path at its starting point, arid a new ordinate w pointing downward in the plane of the ray. On the basis that ray curvature is very small in the atmosphere, some approximations may be made: x = u sin 8 z = z, + U cos 8 1::/« 1. Hence in the uw coordinates the curvature expression becomes 1 &,1 dw 5 (4) r = [I + (%i)'t = du' which combines with (3) to give I dw sin 8 dv (5) r = du 2 = -v- dz . In the following paragraphs the path equation for nearly horizontal rays will be derived, as a function of propagation-velocity gradient dv/dz. As the altitude variation of such rays is smalI, a Taylor expansion of dv/dz is useful: d v , " ( ) 1 ," (z z )2 + (6) dz = V o + Vo Z - Zo + 2vo - 0 ••• where the prime denotes differentiation with respect to z. Substitution of (6) into (5) yields -,dw ---sin 80 ( vo+vo(z-zo)+'" 21vo(z-zo)'" 2) . (7) du Va Here the starting-point values 0 0 , v., are used for the constant ratio (sin O)/v. Following Bertram, one can now include the effects of the earths curvature. For horizontal distances under 1000 km, the equation of the earth's surface is very closely 121 z=-x2/2R, where R is the radius of the earth, In (7L the ray elevation above the surface 2 will thus increase by the additional term x /2R, Secondly, the angle 00 between ray and local vertical will decrease by the amount x/R, Incorporation of these effects into (7) produces an equation valid over the spherical earth: 2 dw sin(Bo - xlR) ( r "x -: x' ) W = v0 v0 + v0 (z + 2R - zo) + 2 (z + 2R zor· (8) This equation must now be re-written in terms of uw coordinates. For nearly horizontal rays, x = U u' (9) and z - Zo + u cas Bo --. 2ro The last term in the z-equation represents the effect of ray curvature: ro is the radius of curvature at u = O. The ro term can be replaced using (5); v~sin 80 2 z = Zo + U cos 80 - ---u . (10) 2vo Further, the sine function in (8) is expanded in a Taylor series, with ~o defined as the complement of 0 0 : x u ~o sin(8 0 - p) = sin Bo - R sin . (11) Equations (9L (10L (11) are substituted into (8) to give the differential equation for the ray path: W ~o) v~ ~) d 2 _ (sin 80 - *' sin [ , " (. .h _ sin 80 2 2 - V0 + v0 u sin '!'o 2 u + 2R du v: Vo 2 (12) V~,(. v~ sin 80 2 u )2] + '2 u sm ~o - 2v u + 2R . o This equation is integrated twice to produce the equation of the ray path in the uw coordinate system. In each coefficient of un, any terms at least 1000 times smaller than their neighbours have been neglected. [~2 v~ ~o ~~v~ v~v~sin v~'sin2~oJ _ sin 80 sin 3 _ 80 4 w - V 2 u + 6 u + 12\2R 2v + --2--; u o o (13) 1 (V~'Sin ~o v~v~'sin ~osin 80) 5 v~' (v~2Sin 280 1 v~sin 80) 6] +----- u+- +------u2 20 2R Zv, 60 4v; 4R 2voR The subscripts "0" refer to values at the origin of the uw coordinates. Typical orders of magnitude for the velocity derivatives are v'Q "'" 100 sec", v~ - 10 m'l sec", and v: _ I m'2sec". 3. Relation belween velocily gradienl and lemperalure profile The path equation (13) requires the derivatives of propagation velocity with respect to elevation. These derivatives may be calculated from a given temperature profile as folIows. A standard result in elementary physics is the equation for pressure in an atmosphere situated in a gravitational field: dp - = - gp(z) (14) dz 122 Temperature (·C) 15 20 z z o o o o o o o'" ~ N N N N N N '"N Density (kg/m') Fig. 1: Choice of co ordinate systems. Fig. 2: Temperature and density profiles: ex­ Abb. 1: Wahl der Koordinatensysteme. ample of strang inversion. Abb. 2: Temperatur- und Dichteprofile: Beispiel einer starken Inversion. where p is the pressure at elevation z, and 9 is the acceleration of gravity (assumed constant over the range of elevations to be encountered).
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